“Nothing comes from nothing,
Nothing ever could.
So, somewhere in my youth or childhood
I must have done something good.”
—Richard Rodgers, “Something Good” from My Fair Lady (1965)
It has become a popular maxim that “absence of evidence is not evidence of absence”. The venerable Quote Investigator traces the phrase back to the nineteenth century, via luminaries such as Martin Rees and Carl Sagan. Yet it is not true. Absence of evidence is evidence of absence—indeed, can be very strong evidence of absence, or even the only evidence of absence. Why is this so? Come, let us reason together.
The proof
At heart, this is a simple question of statistics. If we see a situation in which there is no evidence for A, should this cause us to increase our assessment of the likelihood of the probability that A is not true? We can input some dummy numbers into Bayes' Formula to find out.
Let A be the statement, “The event happened”. Let ¬A be the statement, “The event didn't happen”. Let B be the statement, “There is no evidence that the event happened”. Stipulate the following plausible figures for a situation of almost total ignorance, where we can assume nothing about A and B. Before examining the evidence (or lack thereof), we are totally undecided as to whether the event did or did not happen: P(A)=½ and P(¬A)=½. If the event happened, then there might or might not be any evidence of it (we have no way to know which is more likely): P(B|A)=½. If the event did not happen, then there would be no evidence for it: P(B|¬A)=1. These two mutually exclusive and collectively exhaustive outcomes for B allow us to calculate: P(B)=¾.
Inputting these figures into Bayes' Formula, P(A|B)=½*½/¾=⅓, and P(¬A|B)=½*1/¾=⅔. Even from this position of total ignorance, the mere fact of an absence of evidence has allowed us to move decisively to a position where we are much more inclined to believe ¬A. And even in cases where the likelihood of no evidence of a true event is extremely high, so long as it is higher than zero, Bayes' Formula will grind inexorably on and adjust P(¬A) upwards even if only by a small amount. This should be all that needs saying to convince anyone who has a basic understanding of probability that absence of evidence is evidence of absence. And if you don't possess even that minimal level of statistics knowledge, why are you pronouncing on the validity of statistical statements in the first place?
The error
So how did the false opposite of my maxim gain such credence? The error usually lies in misunderstanding what it means for there to be evidence. Many folks who are reasonably but not highly intelligent—second-order beasts; midwits—tend to think in only linear terms, neglecting probabilities and chaos. In this case, they fail to appreciate the probabilistic nature of knowledge. It is true that absence of evidence is not conclusive proof of absence. But no one said that it was conclusive proof of absence, only evidence of it! Evidence is an inherently probabilistic entity, a term in inductive rather than deductive reasoning, adjusting our priors to varying degrees according to how convincing it is. Absence of evidence does, as proven above, adjust our priors; thus it is evidence, specifically of absence, without needing to be conclusive proof.
The possible exceptions make sense when thought of in terms of our simple Bayesian calculation above. There are hypothetical cases—say, a teapot orbiting the furthest star in the galaxy—where P(B|A) is 1 as well; that is, even if it were true there would be no evidence for us to see. In these cases absence of evidence is not formally evidence of absence, but only because there can be no evidence of anything relating to the question. Whereof one cannot speak, thereof one must be silent
: a good maxim, even if Wittgenstein got a bit carried away in what he felt it applied to. And because these are all silly, pointless hypotheticals, no one is ever actually discussing them when they invoke the false inversion of my maxim.
Cases where one has not yet searched for evidence are also often cited as exceptions to my maxim. But in most cases this does not account for evidence forcing itself upon us! There is in most cases (perhaps in all practical cases) the possibility that evidence could have persisted so visibly that we would not have needed to seek it actively. Even that small quantum of evidence is sufficient for Bayes' Formula to work its magic and adjust P(¬A) upwards.
Indeed, in practice absence of evidence may be the only form of evidence of absence available to us. For recent cases where human testimony is available, it may be that we could get testimony from some near-contemporary source as to the non-existence of the thing—though even that would not be conclusive proof of absence: testimony can be mistaken or even deliberately false. But for cases in the vast billennia of pre-history, what evidence of absence could we find other than simple absence of evidence? Absence of evidence and evidence of absence are, far from being unequal, often nearly synonymous terms.
In short, absence of evidence is indeed evidence of absence. Bayes' Theorem proves that beyond doubt. Absence of evidence is not conclusive proof of absence, but “evidence” is not “conclusive proof”. When we encounter an absence of evidence of a thing, we should proceed in the knowledge that the odds of that thing being true just got shorter.
sum θoətiz abaʊt “absəns ov evidəns biiʸ evidəns ov absəns”
“nuθiŋ kum from nuθiŋ,
nuθiŋᵍ evə kʊd.
soʊ, sumweəʳ in miis yʊʊθ oə caildhʊd
mii must av dʊʊ sumθiŋ gʊd.”
—ricəd rojəz, “sumθiŋ gʊd” from miis feə leidiiʸ (1965)
it av bikum a popyələ maksim ðat “absəns ov evidəns bii not evidəns ov absəns”. ðə venrəbəl kwoʊt investigeitə treis ðə freiz bak tə ðə dein-naθ sencərii, vaiə lʊʊminəriiyiz suc az maətin riis and kaəl seɪɡən. yet it bii not trʊʊ. absəns ov evidəns biiʸ evidəns ov absəns—indiid, kan bii verii stroŋᵍ evidəns ov absəns, oəʳ iivən ðiiʸ oʊnliiʸ evidəns ov absəns. kwes wai bii ðis soʊ? kom, let wii riizən təgeðə.
ðə prʊʊf
at haət, ðis biiʸ a simpəl kweʃcən ov stətistiks. if wii siiʸ a sicʊʊweiʃən in wic ðeə bii noʊʷ evidəns foəʳ A, ʃʊd ðis kʊəz wii tʊʊʷ inkriis wiis asesmənt ov ðə laikliihʊd ov ðə probəbilitii ðat A bii not trʊʊ? wii kan inpʊt sum dumii numbəriz intə beiz-iis fʊəmyələ tə faind aʊt.
kom let A bii ðə steitmənt, “ðiiʸ ivent did hapən”. kom let ¬A bii ðə steitmənt, “ðiiʸ ivent did not hapən”. kom let B bii ðə steitmənt, “ðeə bii noʊʷ evidəns ðat ðiiʸ ivent did hapən”. kom stipyəleit ðə foloʊwiŋ plʊəzibəl figəriz foəʳ a sicʊʊweiʃən ov oəlmoʊst toʊtəl ignərəns, weə wii kan aʃʊʊm nuθiŋᵍ abaʊt A and B. bifʊəʳ egzaminiŋ ðiiʸ evidəns (oə lak ðeərov), wii bii toʊtəliiʸ undisaidəð az tə weðə ðiiʸ ivent did oə did not hapən: P(A)=½ and P(¬A)=½. if ðiiʸ ivent did hapən, ðen ðeə mait oə mait not biiʸ eniiʸ evidəns ov it (wii hav noʊ wei tə noʊ wic bii mʊə laiklii): P(B|A)=½. if ðiiʸ ivent did not hapən, ðen ðeə wʊd bii noʊʷ evidəns foəʳ it: P(B|¬A)=1. ðiiz tʊʊ myʊʊcəliiʸ eksklʊʊsiv and kəlektivliiʸ egzoəstiv aʊtkumiz foə B alaʊ wii tə kalkyəleit: P(B)=¾.
inpʊtiŋ ðiiz figəriz intə beiz-iis fʊəmyələ, P(A|B)=½*½/¾=⅓, and P(¬A|B)=½*1/¾=⅔. iivən from ðis pəziʃən ov toʊtəl ignərəns, ðə miə fakt ov an absəns ov evidəns av allaʊ wii tə mʊʊv disaisvlii tʊʊʷ a pəziʃən weə wii bii muc mʊə inklainəð tə biliiv ¬A. and iivən in keisiz weə ðə laikliihʊd ov noʊʷ evidəns ov a trʊʊʷ ivent biiʸ ekstriimlii hai, soʊ loŋᵍ az it bii haiyə ðan ziəroʊ, beiz-iis fʊəmyələ wil graind ineksərəbliiʸ on and əjust P(¬A) upwəd iivən if oʊnlii baiʸ a smoəl amaʊnt. ðis ʃʊd biiʸ oəl ðat niid seiyiŋ tə kənvins eniiy-um hʊʊ hav a beisik undəstandiŋᵍ ov probəbilitii ðat absəns ov evidəns biiʸ evidəns ov absəns. and if ðii not pəses iivən ðat miniməl levəl ov stətistiks nolij, kwes wai bii ðii prənaʊnsiŋᵍ on ðə vəliditii ov stətistikəl steitməntiz in ðiiʸ umθ pleis?
ðiiʸ erəʳ
soʊ kwes haʊ did gein suc kriidəns ðə fols opəzit ov miis maksim? ðiiʸ erə yʊʊʒəlii laiʸ in misundəstandiŋᵍ wot it miin foə ðeə tə biiʸ evidəns. menii foʊkiz hʊʊ bii riizənəblii but not hailiiʸ intelijənt—tʊʊθ-ʊədə biistiz; mid-witiz—tend tə θink in oʊnlii liniiə tuəmiz, neglektiŋ probəbilitiiyiz and keiyos. in ðis keis, dii feil tʊʊʷ apriiʃiiyeit ðə probəbilisitk neicəʳ ov nolij. it bii trʊʊ ðat absəns ov evidəns bii not kənklʊʊsiv prʊʊf ov absəns. but noʊw-um did sei ðat it did bii kənklʊʊsiv prʊʊf ov absəns, oʊnliiʸ evidəns ov it! evidəns biiʸ an inherəntlii probəbilistik entitiiʸ, a tuəm in induktiv raəðə ðan diduktiv riizəniŋᵍ, əjustiŋᵍ wiis praiəriz tə veəriiyiŋ digriiyiz əkʊədiŋ tə haʊ kənvinsiŋᵍ it biiʸ. absəns ov evidəns dʊʊ, az prʊʊvəð abuv, əjust wiis praiəriz; ðus it biiʸ evidəns, spəsifikliiʸ ov absəns, wiðaʊt niidiŋ tə bii kənklʊʊsiv prʊʊf.
ðə posibəl eksepʃəniz meik sens wen θinkəð ov in tuəmiz ov wiis simpəl beiziiən kalkyəleiʃən abuv. ðeə bii haipəθetikəl keisiz—seiʸ, a tii-pot ʊəbitiŋ ðə fuəðist staəʳ in ðə galəksii—weə P(B|A) biiʸ 1 az wel; ðat biiʸ, iivən if it did bii trʊʊ ðeə wʊd bii noʊʷ evidəns foə wii tə siiʸ. in ðiiz keisiz absəns ov evidəns bii not fʊəməliiʸ evidəns ov absəns, but oʊnlii bikuz ðeə kan bii noʊʷ evidəns ov eniiθiŋᵍ rileitiŋ tə ðə kweʃcən. weərov um not kan spiik, ðeərov um must bii sailənt
: a gʊd maksim, iivən if vitgənʃtain did get a bit kariiyəð aweiʸ in wot hii did fiil it did aplai təəʷ. and bikuz ðiiz bii oəl silii, pointləs haipəθetikəliz, noʊw-um biiʸ evəʳ akcəlii diskusiŋ dii wen diiʸ invoʊk ðə fols invuəʒən ov miis maksim.
keisiz weəʳ um av not yet suəcəð foəʳ evidəns arebiiʸ oəlsoʊ oftən saitəð az eksepʃəniz tə miis maksim. but in moʊst keisiz ðis not əkaʊnt foəʳ evidəns fʊəsiŋᵍ itself upon wii! ðeə biiʸ in moʊst keisiz (pəhaps in oəl praktikəl keisiz) ðə posibilitii ðat evidəns kʊd av pəsist soʊ viziblii ðat wii wʊd not av niid tə siik it aktivliiʸ. iivən ðat smoəl kwontəm ov evidəns bii sufiʃənt foə beiz-iis fʊəmyələ tə wuək itiis majik and əjust P(¬A) upwəd.
indiid, in praktis absəns ov evidəns mei bii ðiiʸ oʊnlii fʊəm ov evidəns ov absəns əveiləbəl tə wii. foə riisənt keisiz weə hyʊʊmən testimənii biiʸ əveiləbəl, it mei bii ðat wii kʊd get testimənii from sum niə-kəntempərərii sʊəs az tə ðə non-egzistəns ov ðə θiŋ—ðoʊʷ iivən ðat wʊd not bii kənklʊʊsiv prʊʊf ov absəns: testimənii kan bii misteikəð oəʳ iivən dilibərətlii fols. but foə keisiz in ðə vast bileniiəmiz ov prii-histərii, kwes wot evidəns ov absəns kʊd wii faind uðə ðan simpəl absəns ov evidəns? absəns ov evidəns and evidəns ov absəns bii, faə from biiyiŋᵍ uniikwəl, oftən niəlii sinoniməs tuəmiz.
in ʃʊət, absəns ov evidəns biiʸ indiid evidəns ov absəns. beiz-iis θiərəm prʊʊv ðat biiyond daʊt. absəns ov evidəns bii not kənkclʊʊsiv prʊʊf ov absəns, but “evidəns” bii not “kənklʊʊsiv prʊʊf”. wen wiiʸ enkaʊntəʳ an absəns ov evidəns ov a θiŋᵍ, wii ʃʊd prəsiid in ðə nolij ðat ðii odiz ov ðat θiŋ biiyiŋ trʊʊ did just get ʃʊətəʳ.